A299773 a(n) is the index of the partition that contains the divisors of n in the list of colexicographically ordered partitions of the sum of the divisors of n.

1, 2, 3, 9, 7, 48, 15, 119, 72, 269, 56, 2740, 101, 1163, 1208, 5218, 297, 24319, 490, 42150, 6669, 14098, 1255, 792335, 5564, 42501, 30585, 432413, 4565, 4513067, 6842, 1251217, 122818, 317297, 124253, 54782479, 21637, 802541, 445414, 48590725, 44583

Offset

1,2

Comments

If n is a noncomposite number (that is, 1 or prime), then a(n) = A000041(n).

For n >= 3, p(sigma(n-2)) < a(n) <= p(sigma(n-1)), where p(n) = A000041(n) and sigma(n) = A000203(n).

Links

Amiram Eldar, Table of n, a(n) for n = 1..3200 (terms 1..700 from Andrew Howroyd)

Example

For n = 4 the sum of the divisors of 4 is 1 + 2 + 4 = 7. Then we have that, in list of colexicographically ordered partitions of 7, the divisors of 4 are in the 9th partition, so a(4) = 9 (see below):
------------------------------------------------------
   k        Diagram        Partitions of 7
------------------------------------------------------
         _ _ _ _ _ _ _
   1    |_| | | | | | |    [1, 1, 1, 1, 1, 1, 1]
   2    |_ _| | | | | |    [2, 1, 1, 1, 1, 1]
   3    |_ _ _| | | | |    [3, 1, 1, 1, 1]
   4    |_ _|   | | | |    [2, 2, 1, 1, 1]
   5    |_ _ _ _| | | |    [4, 1, 1, 1]
   6    |_ _ _|   | | |    [3, 2, 1, 1]
   7    |_ _ _ _ _| | |    [5, 1, 1]
   8    |_ _|   |   | |    [2, 2, 2, 1]
   9    |_ _ _ _|   | |    [4, 2, 1]       <---- Divisors of 4
  10    |_ _ _|     | |    [3, 3, 1]
  11    |_ _ _ _ _ _| |    [6, 1]
  12    |_ _ _|   |   |    [3, 2, 2]
  13    |_ _ _ _ _|   |    [5, 2]
  14    |_ _ _ _|     |    [4, 3]
  15    |_ _ _ _ _ _ _|    [7]
.

Mathematica

b[n_, k_] := b[n, k] = If[k < 1 || k > n, 0, If[n == k, 1, b[n, k + 1] + b[n - k, k]]];
PartIndex[v_] := Module[{s = 1, t = 0}, For[i = Length[v], i >= 1, i--, t += v[[i]]; s += b[t, If[i == 1, 1, v[[i - 1]]]] - b[t, v[[i]]]]; s];
a[n_] := PartIndex[Divisors[n]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 27 2019, after Andrew Howroyd *)

Prog

(PARI) a(n)={my(d=divisors(n)); vecsearch(vecsort(partitions(vecsum(d))), d)} \\ Andrew Howroyd, Jul 15 2018
(PARI) \\ here b(n, k) is A026807.
b(n, k)=polcoeff(1/prod(i=k, n, 1-x^i + O(x*x^n)), n)
PartIndex(v)={my(s=1, t=0); forstep(i=#v, 1, -1, t+=v[i]; s+=b(t, if(i==1, 1, v[i-1])) - b(t, v[i])); s}
a(n)=PartIndex(divisors(n)); \\ Andrew Howroyd, Jul 15 2018

Crossrefs

Cf. A000040, A000041, A000203, A008578, A026807, A027750, A056538, A135010, A141285, A194446, A211992, A272024.

Sequence in context: A205860 A318948 A328424 * A021421 A152812 A246825

Adjacent sequences: A299770 A299771 A299772 * A299774 A299775 A299776

Keyword

nonn

Author

Omar E. Pol, Mar 25 2018

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jul 15 2018

Status

approved